Averages

We know what an average is, but we have two types of averages:

• Time Average
• Ensemble Average

We care about the initial conditions in both cases here. In the ensemble average, we need to make sure that the initial conditions have attenuated before the measurement point. The time average case needs to make sure the initial conditions impact a sufficiently small portion of measurements.

In a nicely behaved (ergodic) system, both averages are the same. This is a system that is:

• Positive recurrent
• Aperiodic
• Irreducible

Note

Irreducibility means a process should be able to get from one state to any other state where the state is number of jobs in the system.

This means the initial state of the system does not matter.

So if we started at 0 jobs or 10, we could still get to any state in the system.

Note

Positive recurrent means that given an irreducible system, any state i is revisited infinitely often, and the time between visits to that state are finite. Therefore, we can define a certain state as being effectively a “restart”. This is when a queue goes to 0.

Note

Aperiodicity is required for the ensemble average to make sense or exist. The state of the system should not be related to time.

Therefore, the time or ensemble average is:
${\lim }_{t\to \infty }\frac{{\sum }_{i=1}^{A(t)}{T}_i}{A(t)}$.

• A(t) is the number of arrivals by time t and $T_i$ is the time in the system of arrival i. The average is taken over one sample path.

The ensemble average is ${\lim }_{t\to \infty }E[T_i]$, where $E[T_i]$ is the average time in the system of job i, where the average is taken over all sample paths.

Summary

Little’s Law says that the average number of jobs in the system equals the product of the average arrival rate into the system and the average time in the system.

For an open system, $E[N]=\lambda E[T]$

For a closed system, with N jobs in process, and X throughput, $N=X\ast E[T]$

If we do have to deal with users and think time, then we care more about the response time $E[R]$. $E[R]=\frac{N}{X}-E[Z]$

Reminder that $E[N]$ is the expected number of customers in the system.

Probabilistic processes are described according to their models, which will be one of:

• Deterministic (D)
• Markov (M)
• General (G)